1. Field of the Invention
This invention relates to a numerical structural analysis system, especially to the numerical structural analysis system to perform the structural analysis of the objective structure for analysis by the load-transfer-path method utilizing the finite-element method.
2. Background Art
In the structural design, high stiffness, high strength, lightweight and so on are aimed. In order to achieve these aims, it is necessary to study all over the structure. It is general to use the stress or strain to study the strength and the stiffness of the structure. However, in the structural design, instead, it is necessary that the conception is designed paying attention to the transfer path of the force or the load transmission. Therefore, it is an important subject to know the transfer path of the force inside of the structure. The transfer of force has been so far estimated by stress. The stress means the force for a unit area. It can be considered that the force is strongly transferred through high stress points. The stress has been analyzed by experimental measurement or computer simulation. It is sometimes performed that the value of the principal stress is shown by an arrow perpendicular to the equal-stress surface, its distribution is drawn in the figure of the structure, and the transfer of force is expressed by the distribution of the arrows.
However, this consideration has some problems. Even if the force is not effectively transferred, a large stress may arise for the sake of stress concentration. The consideration of force transfer with stress may mislead to the wrong conclusion. For example, if there is a small circular hole in the structure, it gives rise to a stress concentration in that part and there arises a large principal stress. But, it is curious that the circular hole is considered to transfer the force strongly. It should not be considered that the part becomes to transfer the large load. If so, such wrong conclusion is conducted that a circular hole should be made in order to transfer the force strongly. Differently, it should be concluded that a circular hole makes to transfer the force a little away from the hole. A new parameter different from stress is necessary in order to consider so.
Therefore, the inventors proposed a new parameter as U* with respect to the force transfer and its path. U* is the parameter based upon the idea entirely different from stress. Though the parameter is conducted according to the intuitive concept, it can be shown that U* is useful in the structural design by clearing the close relation between U* and the force transfer through the example of concrete calculation of U*. Moreover, the method to express definitely the load-transfer path with U* is explained by introducing the U* potential, stiffness line and stiffness decay vector. Hereinafter, the outline of U* is explained. The details of the concrete calculation example and so on are available referring to the non-patent documents 1 through 9.
U* can be obtained according to the ratio of the reactive force with an arbitrary point in the structure unfixed and the reactive force with the point fixed when force is applied to the structure. U* is the parameter to express the connective strength between the loading point and an arbitrary point. It is also a parameter to express the load transfer in the structure. U* is a potential quantity. It is able to show the direction of force transfer in the structure by calculation of U* distribution. It is able to obtain the stiffness decay vector to indicate the load-transfer direction by calculation of the gradient of the U* distribution. The stiffness line to indicate the force transfer path can be derived by tracing the stiffness decay vectors. The path to transfer the load most strongly in the stiffness lines to the supporting point from the loading point is defined as load-transfer path. It becomes possible to estimate the load transfer in the structure quantitatively by using this load-transfer path. That is, using the parameter U*, the aspect of the load transfer in the structure is expressed quantitatively, and the principal load-transfer path inside is found according to the U* distribution in the structure. It is possible to clarify the improvement in the structure by evaluation of the obtained load-transfer path. Optimizing the structure by use of U*, even though only the load transfer is regarded, such solution is obtained as equivalent to the optimum solution in the conventional method where the strain energy is aimed to be uniform. Thus the efficiency of U* is ascertained. And also, other than the computer simulation using FEM, the measurement of U* using the actual vehicle body is performed. This method of structural analysis is decided for calling the load-transfer-path method.
The concept and the definition of U* are explained. FIG. 8 shows the model of arbitrary elastic structure. In FIG. 8(a), the loading point is denoted by A, the supporting point is denoted by B and an arbitrary inner point of the structure is denoted by C. FIG. 8(b) shows the case when the arbitrary point C is constrained. In both of FIGS. 8(a) and 8(b), the same displacement dA is given to the loading point A. In the case of FIG. 8(a), the strain energy stored in the inside is defined as U. In the case of FIG. 8(b), the strain energy stored in the inside is defined as U. The ratio U′/U of both becomes always greater than or equal to 1, this ratio is considered to become as high value as the connectivity between the internal point C and the loading point A is strong. For example, considering one linear spring, the distribution of U′/U with the variant of point C becomes a curve along the spring axis. Instead of U′/U, as shown in the Eq. (1), using the ratio (U′-U)/U′, this value can be expressed as a uniformly decreasing line as it takes 1 at the loading point A and 0 at the supporting point B.
                                                        U              *                        ≡                                          (                                                      U                    ′                                    -                  U                                )                            /                              U                ′                                              =                      1            -                                          (                                                      U                    ′                                    /                  U                                )                                            -                1                                                    ⁢                                                      (        1        )            Eq. (1) is the definition of U*. It is considered to show the strength of connectivity between the loading point A and the point C. U* is a function of the coordinate of the point C.
The load transfer path is explained. The distribution of U* can be obtained by numerical calculation or experiment. When the U* distribution in the structure is expressed by contour, the curve corresponding to ridgeline of the contour is corresponding to the line connecting sequentially the part strongly connected with the loading point. It is considered that the force is transferred following stiff parts. It can be said that the load is transferred along this ridgeline as a path. This ridgeline, the load transfer path S, is shown schematically in FIG. 8(c).
With respect to the load transfer path, three conditions to satisfy for the desirable structure are shown below.
(1) The uniformity of U* distribution in the load-transfer path: The distribution of U* should be close to the ideal line shown in FIG. 9(a).
(2) The continuity of internal stiffness in the load-transfer path: The distribution of U* curvature should be close to the ideal line shown in FIG. 9(b).
(3) The consistency of the load-transfer path: The load-transfer path S1 from the loading point toward the supporting point should coincide with the load-transfer path S2 when the loading point and the supporting point are exchanged each other as shown in FIG. 9(c).
The relation among the structure, load and displacement is explained. Three points A, B and C represent the structure. The relation between the load and the displacement at each point is denoted by Eq. (2).
                              {                                                                      p                  A                                                                                                      p                  B                                                                                                      p                  C                                                              }                =                              [                                                                                K                    AA                                                                                        K                    AB                                                                                        K                    AC                                                                                                                    K                    BA                                                                                        K                    BB                                                                                        K                    BC                                                                                                                    K                    CA                                                                                        K                    CB                                                                                        K                    CC                                                                        ]                    ⁢                      {                                                                                d                    A                                                                                                                    d                    B                                                                                                                    d                    C                                                                        }                                              (        2        )            
Each K with subscripts is the internal stiffness matrix of 3 times 3. p with subscript is the column vector to denote the load. d with subscript is the column vector to denote the displacement. At a glance, it seems like a primitive expression of FEM. But it is not. In Eq. (2), the effect of whole structure with respect to three points is expressed. Besides, even if the load is zero, under the condition to allow the uniform displacement of rigid body, the relation of Eq. (3) and so on is valid.KAA=−(KAB+KAC)  (3)Besides, though K, p, d and so on are matrix or vector, as they cannot be represented by bold type, they are represented with the normal type.
The definition of U* by the Eq. (1) is expressed with Eq. (2). U is the displacement energy when only the supporting point B in FIG. 8(a) is fixed. In FIG. 8(a), as the supporting point B is fixed, dB becomes zero. Hence, using Eq. (2), pA can be expressed as (KAA dA+KAC dC). Using this relation, the displacement energy U in FIG. 8(a) is expressed by Eq. (4).
                                                        U              =                            ⁢                                                (                                      1                    /                    2                                    )                                ⁢                                                      p                    A                                    ·                                      d                    A                                                                                                                                          =                                ⁢                                                      (                                          1                      /                      2                                        )                                    ⁢                                                            (                                                                                                    K                            AA                                                    ⁢                                                      d                            A                                                                          +                                                                              K                            AC                                                    ⁢                                                      d                            C                                                                                              )                                        ·                                          d                      A                                                                                  ⁢                                                                                                      (        4        )            In the productive operation, the representation of vector and tensor is used. The dot symbol means the inner product of vector.
On the other hand, U′ is the displacement energy stored when the arbitrary point C in the structure in FIG. 9(b) is constrained. Also in this case, as the displacement of the loading point A is not changed to keep as dA, the applied force p′A becomes only KAA dA according to Eq. (2). Then, U′ is expressed as Eq. (5).
                              U          ′                =                                            (                              1                /                2                            )                        ⁢                                          p                A                ′                            ·                              d                A                                              =                                    (                              1                /                2                            )                        ⁢                                          (                                                      K                    AA                                    ⁢                                      d                    A                                                  )                            ·                              d                A                                                                        (        5        )            According to Eqs. (4) and (5), Eq. (6) is derived.(U′/U)={(KAAdA)·dA}/{(KAAdA+KACdC)·dA}  (6)The divisor takes constant if dA is given as predetermined value. Then U′/U can be seen to depend upon the numerator KAA. This KAA is −(KAB+KAC) according to Eq. (3). It cannot be considered that it shows only KAC the strength of connectivity between the loading point A and the arbitrary point C. Therefore, the equation is transformed as follows.
According to Eq. (4) and Eq. (5), Eq. (7) is obtained.U′=U−(½)(KACdC)·dA  (7)Therefore, (U′/U) can be expressed as Eq. (8).(U′/U)=1−{(KACdC)·dA}/(2U)  (8)Moreover, using Eq. (8) to the definition of U* in Eq. (1), finally, U* is expressed as Eq. (9).
                                                                        U                *                            =                              1                -                                                      {                                          1                      -                                                                        {                                                                                    (                                                                                                K                                  AC                                                                ⁢                                                                  d                                  C                                                                                            )                                                        ·                                                          d                              A                                                                                }                                                /                                                  (                                                      2                            ⁢                            U                                                    )                                                                                      }                                                        -                    1                                                                                                                          =                                                {                                      1                    -                                                                  (                                                  2                          ⁢                          U                                                )                                            /                                              {                                                                              (                                                                                          K                                AC                                                            ⁢                                                              d                                C                                                                                      )                                                    ·                                                      d                            A                                                                          }                                                                              }                                                  -                  1                                                                                                        =                              {                                  1                  -                                                            (                                                                        p                          A                                                ·                                                  d                          A                                                                    )                                        /                                          {                                                                        (                                                                                    K                              AC                                                        ⁢                                                          d                              C                                                                                )                                                ·                                                  d                          A                                                                    }                                                                      }                                                                        (        9        )            
By expressing as Eq. (9), it has become clear that U* is the quantity depending upon the matrix KAC to express the stiffness between the loading point A and the arbitrary point C. That is, U* at the arbitrary point C can be expressed as the quantity with respect to the strength of connectivity with the loading point A. It could be said that the meaning of the value of U* has been clarified by Eq. (9). Besides, the above is the result based on the model of FIG. 8, when the supporting point are more than or equal to two, moreover, also when certain region is constrained, the similar result can be obtained. In any case, U*, which is defined by the ratio of the energy when the arbitrary point C is fixed and the energy when the arbitrary point C is not fixed, is expressed in the same form as Eq. (9), and it can be expressed by the internal stiffness matrix KAC between the loading point A and the arbitrary point C. Especially, this KAC should be called as the partial stiffness matrix to distinguish from other internal stiffness matrix.
U* potential and the path are explained. This U* can be given at any point in the structure. Considering U* decided at each point as potential, it is called as U* potential. The curve perpendicular to the equal-potential line is called as the stiffness line. The gradient gradU* along the stiffness line is expressing the decaying quantity of stiffness. This decaying quantity is expressed as Eq. (10).k(−)=−gradU*  (10)k(−) is named as the stiffness decaying vector. k(−) is conservative.
The contour of U* and the load path are explained. In general, the ridgeline with respect to the contour of the distribution of scalar quantity is defined as the curve with the smallest gradient among the lines perpendicular to the contour. The curve with the smallest gradient among the stiffness lines, that is, the curve made by connecting the smallest stiffness decay vectors is the ridgeline. If it is considered that the force is transferred along the part where the stiffness decay is small, this ridgeline can be called as the load transfer path. By considering the load transfer path according to the ridgeline of U* distribution, in the case of the presence of a circular hole, it is concluded that the force is transferred along the part a little apart from the circular hole.
The concrete method to calculate U* by definition is explained. In the case to calculate U* strictly as the definition, the following procedure is required.
(1) The energy necessary to cause the displacement at the loading point with the arbitrary point C free.
(2) The necessary energy to give the constant displacement at the loading point A with the arbitrary point C fixed is obtained by the structural calculation such as FEM. The obtained value is substituted into Eq. (1) to calculate U* with respect to the point.(3) The calculation is repeated with varying the arbitrary point C sequentially. Thus, the overall distribution of U* can be obtained.
The finite element method (FEM) is usually employed to obtain U* by calculation. There are two solving methods in FEM, one is the displacement method and another is the force method. The displacement method is used in the general-purpose FEM software. In the displacement method, in the case to calculate repeatedly giving different load at each calculation, that is, in the case to calculate repeatedly with varying the loading condition (mechanical boundary condition), the simultaneous equation with respect to the structural stiffness matrix needs to be solved only once. But, in the case to calculate repeatedly with giving different displacement at each calculation, that is, in the case to calculate repeatedly with varying the displacement condition (geometric boundary condition), the simultaneous equation with respect to the structural stiffness matrix needs to be solved every time repeatedly. Therefore, in the calculation with many repetitions, the calculation complex becomes impractical. In FEM, in one analysis, as almost all the time is spent to calculate the simultaneous equation with respect to the structural stiffness matrix, in the calculation with many repetitions, the calculation complexity becomes impractical.
In the repetitive calculation of U*, the arbitrary point C is constrained. That is, zero displacement is given to the arbitrary point C selected at the different point at every time. As the geometric condition needs to be changed every time as this, the calculation complexity becomes impractical if as it is. The simultaneous linear equation with respect to structural stiffness matrix has even millions of variables. As the equation of millions of elements is to be solved, it takes a long time to solve it only once. Moreover, in order to find the distribution of U*, it is repeated millions of times. It is very important to reduce those repetition times.
The matrix KAC instead of U* is explained to have only to be calculated. KAC can be considered as the matrix of spring constant between the loading point of A and the arbitrary point of C. U* is given by the equation (1).U*=(1−(U′/U))  (1)Eq. (1) can be expressed as Eq. (9) also.U*={1−(2U)/{(KACdC)·dA}}−1  (9)Here, U is the deformation energy when the arbitrary point C is not fixed. The displacement dC and the displacement dA are the displacement of the arbitrary point C and the displacement of the loading point A respectively when the arbitrary point C is not fixed. Therefore, U, dC and dA have only to be calculated once at the first time. In order to calculate U* from Eq. (9), the matrix KAC is only necessary to be calculated on each point. That is, the calculation necessary to obtain U* is just the calculation to obtain the matrix KAC on each point repeatedly.
Here is explained what the matrix KAC is. Two kinds of methods to express the load and the displacement of the structure are described for the explanation. First, the expressing method of the structure with structural stiffness matrix is explained. The structure is expressed with dividing to mesh elements corresponding to finite elements of FEM. The number of the elements may extend to one million in some example. Each node point of these elements is numbered sequentially and the force and the displacement applied thereon are represented by p and d with subscripts. Both of them are vectors. Here, only such case as the force is directly proportional to the displacement should be treated. That is, the case where the displacement is in linear relation with the force should be treated. This is expressed with Eq. (11).
                              {                                                                      p                  1                                                                                                      p                  2                                                                                    ⋮                                                                                      p                  A                                                                                    ⋮                                                                                      p                  B                                                                                    ⋮                                                                                      p                  C                                                                                    ⋮                                                                                      p                  n                                                              }                =                              [                                                                                (                    11                    )                                                                                        (                    12                    )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (                    21                    )                                                                                        (                    22                    )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (                    AA                    )                                                                                        (                    AB                    )                                                                                        (                                          A                      ⁢                      C                                        )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (                    BA                    )                                                                                        (                    BB                    )                                                                                        (                    BC                    )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (                    CA                    )                                                                                        (                    CB                    )                                                                                        (                    CC                    )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (                    nn                    )                                                                        ]                    ⁢                      {                                                                                d                    1                                                                                                                    d                    2                                                                                                ⋮                                                                                                  d                    A                                                                                                ⋮                                                                                                  d                    B                                                                                                ⋮                                                                                                  d                    C                                                                                                ⋮                                                                                                  d                    n                                                                        }                                              (        11        )            
The other elements of the matrix are omitted in the equation. As the number n of the nodes sometimes becomes extremely large, the number n*n of the elements of this matrix becomes very large. Moreover, p and d are both vectors. Each of them is composed of three scalars. One element of the matrix in Eq. (11) is composed of 3*3 scalars. So, actually, it is composed of even more elements. This matrix is well known as the stiffness matrix in FEM. It is the main calculation in FEM to solve this simultaneous equation considering the displacement of rigid body. As the number of the elements is large, this calculation takes a much of time. As usual, it is calculated with general-purpose calculation software. The first expression method is to express the relation between the load and the displacement of the structure with this stiffness matrix.
Next, here is explained the method to express the structure with only three points in the case to give the loading point A, the supporting point B and the arbitrary point C. In the case that a certain structure is expressed with only three points A, B and C with respect the arbitrary point C, the mesh of FEM becomes disrelated. It becomes the relation between the load and the displacement with respect to only tree points. The equation of the relation is simple and it is expressed with above Eq. (2).
K with subscript in this expression is called the internal stiffness matrix. The element of this matrix is 3*3 as seen in Eq. (2). Besides, its one element is also composed of 3*3 scalars. That is, the matrix of Eq. (2) is composed of 9*9 scalars. The second expression method is to express the relation between the load and the displacement of the structure with this internal stiffness matrix. This KAC in Eq. (2) is KAC in Eq. (9). That is, if this KAC is obtained, U* will be obtained by Eq. (9). The detail of the above explanation could be obtained by referring to the following reports on U* with abstracts by the inventors of this invention.
In the non-patent document 1, Conditions for desirable structures based on a concept of load transfer courses is reported. A new concept of a parameter U* is introduced to express load transfer courses for a whole structure. A degree of connection between a loading point and an internal arbitrary point in the structure can be quantitatively expressed with the parameter U*. Based on the proposed concept, three conditions for desirable structures are introduced: (1) Continuity of U*, (2) Linearity of U*, (3) Consistency of courses. After introducing these three conditions as objective functions, structural optimization with numerical computation is carried out. Despite the fact that no concept of stresses or strains is introduced, the obtained structure has a reasonable shape. Finally, the load transfer courses for a simple structure are experimentally measured and these values demonstrate that the parameter U* can effectively be used.
In the non-patent document 2, Vibration Reduction for Cabins of Heavy-Duty Trucks with a Concept of Load Path is reported. The load transfer paths in the cabin structures of heavy-duty trucks are investigated under static loading and the results are applied to the vibration reduction of cabins. In a preliminary simulation using a simple model, it is shown that the floor panel vibration is closely related to the stiffness of the front cross-member of the floor structure. Load path analyses using the finite element method show that the load paths have some discontinuities and non-uniformities in the front cross-member which cause the low stiffness of the member.
In the non-patent document 3, Application of ADAMS for Vibration Analysis and Structure Evaluation By NASTRAN for Cab Floor of Heavy-Duty Truck is reported. The load transfer paths in the cabin structures of heavy-duty trucks are investigated under static loading, and the results are applied to the reduction of vibration in cabins. In a preliminary simulation using a simple model with ADAMS/Vibration, it is shown that vibration in the floor panel is closely related to the stiffness of the front cross-member of the floor structure. Load path analyses using the finite element method with NASTRAN show that the load paths have some discontinuities and non-uniformities in the front cross-member, reducing that member's stiffness.
In the non-patent document 4, ADAMS application for the floor vibration in the cabin of heavy-duty trucks and U* analysis of the load path by NASTRAN is reported. Realization of lightweight, cost-effective structures of heavy-duty trucks is an important aspect of structural designs, and numerical analyses have played a key role in this regard. In a preliminary simulation using ADAMS/Vibration, it is shown that the floor panel vibration is closely related to the stiffness of the front cross-member. Load path analyses using MSC/NASTRAN show that the load transfer paths have some discontinuities and non-uniformities in the front cross-member.
In the non-patent document 5, Expression of Load Transfer Paths in Structural Analysis and its Applications is reported. A new parameter U* is introduced to express the load transfer path using FEM. As an example, load path U* analysis is applied to a plate structure with a circular hole. Although the effect of stress concentration suggests strong force transfer at the corner of the hole, the obtained position of the load transfer path avoids a corner of the hole. This result coincides with their intuitive prediction. Moreover, they try to extend the calculation method of U* analysis to a structure with more complex boundary conditions. The effectiveness of the introduced method is verified using the FEM model of an actual heavy-duty track cab.
In the non-patent document 6, Load Path Optimization and U* Structural Analysis for Passenger Car Compartments under Frontal Collision is reported. A new concept, a parameter U*, is introduced to express load transfer in a structure. Two cases of U* analysis for a floor structure of a passenger compartment are examined. In the first case, three conditions of U* are introduced as objective functions, and GA structural optimization is applied. The emergent floor structure after the GA calculation has a unique shape in which a member connects the frontal part of an under-floor member and the rear part of a side-sill. In the second case, the U* values and the load paths in a floor structure under collision are calculated by use of PAM-CRASH. As the collision progresses, the under-floor member becomes the principal load path, and in the final stage of the collision the roll of the under-floor member becomes dominant.
In the non-patent document 7, Vibration reduction in the cabins of heavy-duty trucks using the theory of load transfer paths is reported. The objective of this study is to investigate the load transfer paths in the cabin structures of heavy-duty trucks under static loading, and to apply the results to reduce vibration in cabins. In a preliminary simulation using a simple model, it is shown that the floor panel vibration is closely related to the stiffness of the front cross-member of the floor structure. Load path U* analyses using the finite element method show that the low stiffness of the front cross-member is caused by discontinuities and non-uniformities in the load paths.
In the non-patent document 8, Expression of Load Transfer Paths in Structures is reported. A concept of a parameter U* has been introduced by the authors to express load transfer paths in a structure. In this paper, matrix formulation of internal stiffness shows that the value of U* expresses a degree of connection between a loading point and an internal arbitrary point. Stiffness fields, stiffness lines, and stiffness decay vectors are defined using newly introduced U* potential lines. A concept of a load path can be expressed as a stiffness line that has a minimum stiffness decay vector. A simple model structure is calculated using FEM for an application of U* analysis. The distribution of U* values shows that a diagonal member between a loading point and a supporting point plays an important role for the load transfer.
In the non-patent document 9, Experimental study of U* analysis in load transfer using the actual heavy-duty track cabin structure and scaled model is reported. The distribution of U* is known to represent the load transfer path in the structure. Two experimental measuring method of U* is developed of U* with respect to the actual heavy-duty truck cabin structure and the scaled plastic model. In these methods, different from the conventional method, the stiffness data of each member is not necessary. In FEM, the effect of the actual plate to play the important role in U* analysis cannot be expressed. By using the plastic scaled model, the strengthening effect can be directly measured according to the distribution of U* value.    Non-patent document 1: Kunihiro Takahashi; Conditions for desirable structures based on a concept of load transfer courses, Proc. ISEC-01, pp. 699-702, 2001.    Non-patent document 2: Toshiaki Sakurai, Hiroaki Hoshino, Kunihiro Takahashi; Vibration Reduction for Cabins of Heavy-Duty Trucks with a Concept of Load Path, Proc. JSAE No. 36-02, pp. 5-8, 2002 (in Japanese with English summary).    Non-patent document 3: Hiroaki Hoshino, Toshiaki Sakurai, Kunihiro Takahashi; Application of ADAMS for Vibration Analysis and Structure Evaluation By NASTRAN for Cab Floor of Heavy-Duty Truck, The 1st MSC.ADAMS European User Conference, London, November, 2002.    Non-patent document 4: Toshiaki Sakurai, Hiroaki Hoshino, Masatoshi Abe, Kunihiro Takahashi; ADAMS Application for the Floor Vibration in the Cabin of Heavy-duty Trucks and U* Analysis of the Load Path by NASTRAN, (MSC.ADAMS User Conference 2002).    Non-patent document 5: Toshiaki Sakurai, Masatoshi Abe, Soei Okina, Kunihiro Takahashi; Expression of Load Transfer Paths in Structural Analysis and its Applications, Trans. JSCES, Vol. 8, pp. 401-404, May 2003.    Non-patent document 6: Toshiaki Sakurai, Junichi Tanaka, Akinori Otani, Changjun Zhang, Kunihiro Takahashi; Load Path Optimization and U* Structural Analysis for Passenger Car Compartments under Frontal Collision, International Body Engineering Conference 2003, pp. 181-186, JSAE 20037007, SAE 2003-01-2734, 2003.    Non-patent document 7: Hiroaki Hoshino, Toshiaki Sakurai, Kunihiro Takahashi: Vibration reduction in the cabins of heavy-duty trucks using the theory of load transfer paths, JSAE Review 24 (2003) 165-171.    Non-patent document 8: Kunihiro Takahashi, Toshiaki Sakurai: Expression of Load Transfer Paths in Structures, J. JSME, (A)71-708 (2005), pp. 1097-1102.    Non-patent document 9: Kengo Inoue, Yuichiro Ichiki, Ikuma Matsuda, Toshiaki Sakurai, Hideaki Ishii, Tetsuo Nohara, Hiroaki Hoshino, Kunihiro Takahashi: Experimental study of U* analysis in load transfer using the actual heavy-duty track cabin structure and scaled model, Proc. JSAE, No. 90-04, pp. 27-30, 2004.